Question

How many way sit 4man and 3women in a row thus 2 women can't sit together?

Answer 1

In how many ways can 4 men and 3 women be arranged in a round table:  i) if the women always sit together? ii) if the women never sit together?



Case 1: if the women always sit together

We can club the 3 women in one group and the number of arrangements of this group will be 3!3!

Now, 4 men and 1 group of women can be arranged arround a round table =(5–1)!=(5–1)!

Hence, Total arrangements =4!3!=144=4!3!=144

Case 2: if all 3 women never sit together

Total arrangements without any constraints =(4+3–1)!=(4+3–1)!

From case 1 we know the arrangements if women always sit together

Hence, Total arrangements =6!−4!3!=576=6!−4!3!=576

Case 3: if no 2 women sit together

Lets make a group of 2 Men and 2 Women such that the women are between the 2 men and find their arrangements. This can be done in 4C2×3C2×2!×2!=724C2×3C2×2!×2!=72

Now the above group and rest of the 2 men and 1 woman can be arrange around a table in (4–1)!=6(4–1)!=6 ways

∴∴ Arrangements when 2 women sits together =72×6=432=72×6=432

We also need to remove cases where all 3 women sit together

Therefore Total number of arrangements =6!−4!3!−72×6=144